Preschool math lessons:

A developmental guide for the science-minded parent

Preschool math lessons don't have to be tedious. They don't even have to focus on counting.

Here I review the highlights of research about early mathematics education, along with
specific tips for creating mathematical experiences that are playful and
stimulating. I also present some
research-inspired number activities
and
math games
designed to help preschoolers hone their number sense.

The payoff? By sharing these experiences, you get to watch your
child’s mind at work. And your child gets to learn about numbers through
play and exploration.

Such improvements may bode well for future achievement. An
analysis tracking thousands of English-speaking students found that kids who entered kindergarten with a strong grasp of

counting

relative magnitudes, and

ordinality (sequencing magnitudes)

achieved better math scores in the later years.

Moreover, these preschool math skills were more predictive of general scholastic achievement than were language, attention, or social skills (Duncan et al 2007).

That doesn’t mean that preschool math lessons cause better
scholastic achievement. But it suggests that preschool math ability is
an important bellwether of a child’s scholastic preparation.

So let’s get down to business. How can we create preschool math lessons that are stimulating for kids and their teachers?

Understand that number sense is distinct from a knowledge of counting

Preschool math lessons often emphasize counting, and for good reason.
Understanding the counting system is a prerequisite for learning about
the number line. It also helps kids grasp the concept of addition.

But it’s important to realize that number sense doesn’t depend on language.

In a way, this shouldn't surprise us. After all, even nonhuman animals have reason to keep track of quantity.

Which tree bears more fruit?

If three predators hide behind a rock, and one walks away, how many are left?

Studies show that all sorts of non-linguistic animals-—birds, rats, monkeys—-can solve such problems (Dehaene 1999).

And research also suggests that human children can perform basic mathematical tasks without knowledge of number words.

Case in point: Recent research conducted by cognitive
neuroscientists on kids who speak only Walpiri or Anindilyakwa, two
native Australian languages (Butterworth et al 2008).

These languages include number words for only three
numerosities--“one," “two," and any imprecise quantity that is "more
than two."

Yet 4- to 7-year old speakers of these languages performed as well or better than English speakers when they were asked to

• briefly examine a small set of tokens and then assemble an identical set of tokens from memory

• listen to a series of up to 7 taps and then place the corresponding number of tokens on a mat

• spontaneously subdivide a set of 6 or 9 items into three equal
sets when they were told to “share" these items among three toy bears

• briefly observe two small sets of tokens and assemble a third set of tokens that represented the sum

Again, these kids knew no words for specifying precise quantities of more than two.

The implications?

Kids can learn a lot about numbers without knowing how to count.
Preschool math lessons can extend to activities that don’t require
counting at all.

In fact, research suggests that kids need to learn a lot about
numbers before they begin to label specific quantities with counting
words. We can use this information to design a variety of interesting
preschool math lessons.

Help your child develop a strong sense of “numerosity" and other basic math concepts

Numerosity

How many cats are in your backyard? How many hours are left before bedtime?

Psychologists use the term “numerosity" to denote the number of things in a set.

Numerosity is the conceptual bedrock for most basic math skills.
Kids who don’t grasp numerosity—as an abstract concept and as an
intuition about the meaning of specific magnitudes—have trouble
understanding the counting system. They also have more difficulty with
arithmetic and making measurements (Booth and Siegler 2006; Siegler and
Booth 2004).

How can we help kids learn about numerosity? Recent experiments on American first graders suggest that approximation activities are helpful. Show kids two sets of objects and ask them to judge which is bigger -- without counting (Hyde et al 2104).

Such activities might help preschoolers, too. But very young children have a lot more to learn. They need to know that specific numbers signify, or map onto, specific quantities. And this understanding may improve with practical, hands-on experience with real sets (Dehaene 1999; Hirsh-Pasek et al 2003).

Consider, for instance, how everyday games can help children learn these crucial concepts (Butterworth 1999):

The “one-to-one principle." How do you know if two sets
have the same numerosity? Your child can check by matching up the
members of each set in one-to-one correspondence. If they can do this
with nothing left over, the numerosity is the same. How many? Help you
child count when you’re done. You can also pose a simpler, but related
task to your child: Have a tea party in which each attendee (human,
stuffed animal or action figure) gets one of each item—plate, cup,
spoon, cookie, etc.

Numerosity can apply to sets of anything. You can drive this point home by having kids work
with a variety of objects as well as intangible things, like the number
of times they hear you clap.

The numerosity of a set can be changed by addition or subtraction.
Games like Hi Ho Cherry-O help kids put this concept into practice.
What happens to your tree of cherries when you add one cherry? Or take
one away?

Counting

Counting skills depend on a child's grasp of numerosity, including an understanding that (1) each number word picks out a specific numerosity, and (2) each item to be counted is counted once and only once. In addition, as cognitive psychologists Rochel Gelman and Randy Gallistel have noted (1978) kids also need to learn:

• That number words must be recited in the same order (aka the “stability principle")

• That later number words in the counting sequence refer to bigger numerosities

• That the last word counted represents the numerosity of the set (aka the “cardinality principle")

Experimental studies suggest that 14-month old infants can keep
precise track of quantities up to 3—remembering, for instance, if a box
contains 1, 2, or 3 balls (Feigenson and Carey 2003).

Babies also know something about the approximate, relative value of different numbers.
Show babies a series of visual displays—each featuring an array of
dots—and their brains will respond differently depending on what they
see. A baby who is used to seeing displays of 4 dots will perk up when
you show her a display of 8 dots (Izard et al 2008).

There are limits to these abilities. For instance, babies don’t
understand the meaning of counting words. And babies don’t make fine
distinctions between number sets. Ten-month old babies, who can
distinguish between sets of 8 and 12 objects, do not discriminate between sets of 8 and 10 objects (Xu and Arriaga 2007).

But the important point is this: By the time your child is 2-3 years old, he already knows
something about tracking very small numbers, and he understands
something about “greater than" and “less than."

These findings indicate a good starting point for preschool math lessons:

The first three numbers. Focus on learning the words for numbers 1-3 and you’ll be working with your child’s pre-existing number sense.

Relative magnitudes. You can help kids sharpen their
intuitive sense of “how much" a number represents by encouraging them to
compare and arrange sets of objects in order of relative magnitude.

Be patient

It’s one thing to keep track of three objects, another to understand
that the number term “3" refers to all sets of three things. If your
child is just starting to learn about numbers, expect slow progress.

Research suggests that a 2- or 3-year old kid who has learned the
meaning of “1" will take another six months to learn about “2" and
three months beyond that to learn about “3."

Altogether, it can take kids about a year to really understand how the counting system works (Wynn 1992).

Be ready for a faster pace after your child has mastered numbers up to 4

Once your child really “gets" the first four numbers, he will probably find it much easier to tackle higher numbers (Wynn 1992).

Don’t push. Learning should be spontaneous and fun.

Granted, there is some debate about whether or not it’s a good idea
to push an academic curriculum on young children. At present, I can’t
find any experimental studies addressing the question. So the jury is
still out.

But I think it’s significant that the vast majority of human
societies don’t expect children to sit still for formal education until
they are between 5 and 7 years old. This may reflect a universal trend
of brain development. The frontal lobes—-which permit us to reflect,
reason, and control our impulses-—don’t begin to mature until children
are around 5-6 years old (Eliot 2000).

Formal instruction before this age may therefore be an exercise
in futility, or at least frustration. And even if kids learn something
this way, is it worth it?

Some researchers are concerned that an overly-regimented approach
to early childhood education could backfire, making restless young
children develop negative attitudes about school (Blakemore and Frith
2005; Diamond and Hopson 1999).

As I've written elsewhere,
some school kids are developing math anxiety as early as the first
grade. The results can be intellectually crippling, because anxious kids
are more likely to avoid math practice and fall behind.

Given these concerns, I think it makes good sense to play it safe
and avoid preschool math lessons that feel like lectures or drills.
Learning about math should be fun, and—-ideally-—should reflect your
child’s own spontaneous interest.

Don't assume that school is the best place to learn about math

Some schools might do a great job presenting preschool math lessons.

But a recent British study suggests that even high-quality preschools may not foster long-term achievement in math.

According to this research,
the better predictor of long-term math achievement is the quality of a preschooler's home learning environment.

Help unfocused, distractable kids prepare for kindergarten math by improving their self-control

A recent study tracking 228 American children reports a link between
early math skills and self-control. Three-year-olds who scored low on
executive control -- the ability to regulate one's impulses and
attention -- had poorer math skills in kindergarten (Clark et al 2012).

Does this mean that every preschooler with weak executive control
is headed for trouble with mathematics? No. But given the evidence that
we can help kids develop better self-control, it seems a good idea to
identify struggling kids and invest some effort in boosting their
executive control. For more information, check out my evidence-based article about teaching self control.